Despite regulatory reforms, Money Market Funds (MMFs) experienced severe stress in March 2020, following large redemptions and dislocations in short-term markets. We provide a model showing the trade-offs between liquidity and capital preservation services offered by MMFs. We show that in a crisis, MMFs cannot provide liquidity and capital preservation to investors at the same time. As a result, investors have an incentive to run pre-emptively. We calibrate our model on data from USD MMFs and find that most funds would have been unable to meet redemptions above 30% mid-March 2020. Unless short-term funding markets are made resilient in times of stress, MMFs will face similar challenges during future liquidity crises.